Difference between revisions of "Mock AIME II 2012 Problems/Problem 3"
(Created page with "==Problem== Let <math>\{a_n\}</math> be a recursion defined such that <math>a_1=1, a_2=20</math>, and <math>a_n=\sqrt{\left| a_{n-1}^2-a_{n-2}^2 \right|}</math> where <math>n\ge ...") |
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==Problem== | ==Problem== | ||
− | + | The <math>\textit{digital root}</math> of a number is defined as the result obtained by repeatedly adding the digits of the number until a single digit remains. For example, the <math>\textit{digital root}</math> of <math>237</math> is <math>3</math> (<math>2+3+7=12, 1+2=3</math>). Find the <math>\textit{digital root}</math> of <math>2012^{2012^{2012}}</math>. | |
==Solution== | ==Solution== |
Latest revision as of 01:02, 11 July 2012
Problem
The of a number is defined as the result obtained by repeatedly adding the digits of the number until a single digit remains. For example, the
of
is
(
). Find the
of
.
Solution
The sum of the digits when we get this down to a one digit number is the same thing as the number modulo 9. This is because let’s say we start with a number . When we take the sum of the digits modulo 9, this is the same thing as taking the number modulo 9. However, the sum of the digits modulo 9 is the same thing as the sum of the sum of the digits modulo 9, and this pattern repeats.
We are left with finding . Note that
, therefore we get
. By Euler’s Totient Theorem,
or
. We now need to find
, which is equal to
. Note that
, and
, so this pattern repeats
in the exponents. Since
, we get
. From this, we get
, so we need to find
. Since
, we get an answer of
.